The semiclassical CLR inequality on \(\mathbb {R}^n\) , \(n\ge 3\) , is the main tool to extend the semi-classical Weyl's law for Schrödinger operators \(h^2\Delta +V\)  with smooth potentials \(V\)  to Schrödinger operators with potentials in \(L_{n/2}(\mathbb {R}^n)\)  (see [1]}). This result can be traced back to the work of Birman and his collaborators in the late 60s and early 70s (see [2]}, [3]}).
Likewise, we can extend the semiclassical Weyl's law for fractional Schrödinger operators \(h^{\frac{n}{p}}\Delta ^{\frac{n}{2p}}+V\)  for smooth potentials to potentials in a suitable \(L_q\) -class (see [4]} and the references therein).
